The Bieri-Neumann-Strebel sets of quasi-projective groups

Abstract

Let $X$ be a smooth complex quasi-projective variety and $\Gamma=\pi_1(X)$. Let $\chi \colon \Gamma \to \mathbb{R}$ be an additive character. We prove that the ray $[\chi]$ does not belong to the BNS set $\Sigma(\Gamma)$ if and only if it comes as a pullback along an algebraic fibration $f \colon X \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. We also prove that if $\pi_1(X)$ admits a solvable quotient which is not virtually nilpotent, there exists a finite \'etale cover $X_1 \to X$ and a fibration $f \colon X_1 \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. Both of these results were proved by Delzant in the case when $X$ is a compact K\"ahler manifold. We deduce that $\Gamma$ is virtually solvable if and only if it is virtually nilpotent, generalising the theorems of Delzant and Arapura-Nori. As a byproduct, we prove a version of Simpson's Lefschetz Theorem for the integral leaves of logarithmic $1$-forms that do not extend to any partial compactification. We give two applications of our results. First, we strengthen the recent theorem of Cadorel-Deng-Yamanoi on virtual nilpotency of fundamental groups of quasi-projective $h$-special and weakly special manifolds. Second, we prove the sharpness of Suciu's tropical bound for the fundamental groups of smooth quasi-projective varieties and answer a question of Suciu on the topology of hyperplane arrangements.